Symmetrize bone generates flipped X axis
Yes this is how it is supposed to be. And you are right, you cannot easily copy over your limit constraints, if they are defined in world space. Since the "world axis" are not mirrored. 

But if you set them to local space, they should copy just fine. And local space is what you want anyway in most rig constraints. Otherwise depending on rotation your bend axis would change and a bend forward in t-pose would become a painful bend in any other pose ;)